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Lotto Urn with a Twist.

Q: An urn contains lotto tickets numbered 1 to 100. You play the game by paying a fee. The rules of the game are as follows, you need to draw two tickets, you are paid the value of the lower number of the tickets. What is a fair price to play this game?

A: The first ticket drawn could be any of the tickets from 1 to 100. The second draw sets up the payoff which will be the lower of the two. Assume the first drawn ticket has the number \(i\) on it. The second ticket drawn could be either greater or lesser with probabilitie as shown in the figure below.

If the chosen number is lesser, the expected value is \(\frac{i-1}{2}\). Likewise, if the chosen value is greater the expected value is \(\frac{N-i}{2}\). Thus, the total expected value can be computed as
$$
E = \frac{i-1}{N}\times \frac{i-1}{2} + \frac{N-i}{N}\times\frac{N-i}{2}
$$
Note, the above expression is just for the case when \(i^{th}\) ticket is selected. This happens with probability \(\frac{1}{N}\). So, the resulting expected value is
$$
E = \frac{1}{N}\sum_{i=1}^{N}\frac{(i-1)^{2}}{2N} + \frac{(N-i)^{2}}{2N}
$$
Using the identities for the sum of integers to \(n\) and the sum of squares of integers to \(n^{2}\), the above expression can be simplified to
$$
\frac{n(n+1)(2n+1)}{3} - n(n+1)^{2} + n + n^{3}
$$
Note the above expression is only valid for large \(N\). As we have already selected one ticket there remains only \(N-1\) tickets, but we can ignore this change for now.
Plugging in \(n = 100\) yields the expected payoff to be \(\approx 32.5\) which is the breakeven fee to play this game.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.